Solving Inequalities Using Four Basic Operations
Solving Inequality Using Four Basic Operations
Here, we’re going to look at a slightly more complex inequality. Recall the rules for solving any inequality are very similar to the rules for solving an equation. You can manipulate the inequality by performing operations, so long as you perform the same operations to both sides of the inequality.
There is, however, one additional provision for inequalities. If you multiply or divide both sides of the inequality by a negative number, you have to reverse the sign on the inequality. For instance, if I were to multiply both sides of this inequality by -1, I would have to change this less than sign to a greater than sign.
Let’s keep that in mind as we solve this. Now, the first thing we want to do here is eliminate the denominator on this fraction, so that we can just have whole numbers and whole number coefficients. What we’re going to do is simplify the denominator and then multiply everything through by the denominator.
What we have here is 8 minus 12, which is equivalent to -4. We need to multiply each term in this inequality by -4. We can multiply- if we multiply this fraction by -4, we just eliminate the denominator. What we’re left with is 5x plus 2. Multiply 3x by -4 and we get -12x Minus 12x. We multiply 10x by -4 and we get -40x.
We multiply -1 by -4 and we get a positive 4. Note that we multiplied by a negative number, so we have to reverse the sign on the inequality. This is now going to become a greater than sign. Now, all we have to do is isolate the x on one side of the equation and solve for it. We need to add 40x to both sides.
That’ll eliminate x from the right side. We’ll subtract 2 from both sides. That will get rid of the twos and get rid of the 40x’s. Now we just need to combine our x’s. We have 5 minus 12 plus 40, which is going to be 33x, which is greater than 4 minus 2 is 2. We know that x is greater than 2/33. That is going to be our solution to this inequality.
Let’s make sure by plugging back in a value for x that is greater than 2/33 just to make sure that this inequality is correct. 1 is greater than 2/33, so we’ll plug in 1. What we’ve got here, if we plug in 1, is we’ve got 5 plus 2 over 8 minus 12 plus 3 is less than 10 minus 1. We’ll simplify this. 7/-4 plus 3 is less than 9. 7/-4 is -1 3/4 plus 3 is going to be positive 1 1/4. 1 1/4 is less than 9. This inequality holds correct.